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An $f$-ring $A$ is an if for every minimal prime $\ell$-ideal $P$ of $A$, $A/P$ is a valuation domain. A topological space $X$ is an if $C\left(X\right)$ is an SV $f$-ring. SV $f$-rings and spaces were introduced in [], []. Since then a number of articles on SV $f$-rings and spaces and on related $f$-rings and spaces have appeared. This article surveys what is known about these $f$-rings and spaces and introduces a number of new results that help to clarify the relationship between SV $f$-rings and spaces and related $f$-rings and spaces.

A function $f$ mapping the topological space $X$ to the space $Y$ is called a function if for every cozeroset neighborhood $H$ of a zeroset $Z$ in $X$, the image $f\left(H\right)$ is a neighborhood of ${cl}_Y\left(f\left(Z\right)\right)$ in $Y$. We say $f$ has the if whenever $U$, $V$ are cozerosets and $Z$ is a zeroset of $X$ such that $U\subseteq Z\subseteq V$, there is a zeroset $Z^{\text{'}}$ of $Y$ such that $f\left(U\right)\subseteq Z^{\text{'}}\subseteq f\left(V\right)$. A surjective function is z-open if and only if it maps cozerosets to cozerosets and has the z-separation property. We investigate z-open functions and other functions that map cozerosets to cozerosets....

A lattice-ordered ring $ℝ$ is called an if each of its order ideals is a ring ideal. Generalizing earlier work of Basly and Triki, OIRI-rings are characterized as those $f$-rings $ℝ$ such that $ℝ/𝕀$ is contained in an $f$-ring with an identity element that is a strong order unit for some nil $l$-ideal $𝕀$ of $ℝ$. In particular, if $P\left(ℝ\right)$ denotes the set of nilpotent elements of the $f$-ring $ℝ$, then $ℝ$ is an OIRI-ring if and only if $ℝ/P\left(ℝ\right)$ is contained in an $f$-ring with an identity element that is a strong order unit.